| Description |
1 online resource (xiii, 401 pages) : illustrations |
| Contents |
Part 1 Metric Spaces -- 1 Calculus Review 3 -- Real Numbers 3 -- Limits and Continuity 14 -- 2 Countable and Uncountable Sets 18 -- Equivalence and Cardinality 18 -- Cantor Set 25 -- Monotone Functions 31 -- 3 Metrics and Norms 36 -- Metric Spaces 37 -- Normed Vector Spaces 39 -- More Inequalities 43 -- Limits in Metric Spaces 45 -- 4 Open Sets and Closed Sets 51 -- Open Sets 51 -- Closed Sets 53 -- Relative Metric 60 -- 5 Continuity 63 -- Continuous Functions 63 -- Homeomorphisms 69 -- Space of Continuous Functions 73 -- 6 Connectedness 78 -- Connected Sets 78 -- 7 Completeness 89 -- Totally Bounded Sets 89 -- Complete Metric Spaces 92 -- Fixed Points 97 -- Completions 102 -- 8 Compactness 108 -- Compact Metric Spaces 108 -- Uniform Continuity 114 -- Equivalent Metrics 120 -- 9 Category 128 -- Discontinuous Functions 128 -- Baire Category Theorem 131 -- Part 2 Function Spaces -- 10 Sequences of Functions 139 -- Historical Background 139 -- Pointwise and Uniform Convergence 143 -- Interchanging Limits 150 -- Space of Bounded Functions 153 -- 11 Space of Continuous Functions 162 -- Weierstrass Theorem 162 -- Trigonometric Polynomials 170 -- Infinitely Differentiable Functions 176 -- Equicontinuity 178 -- Continuity and Category 183 -- 12 Stone-Weierstrass Theorem 188 -- Algebras and Lattices 188 -- Stone-Weierstrass Theorem 194 -- 13 Functions of Bounded Variation 202 -- Functions of Bounded Variation 202 -- Helly's First Theorem 210 -- 14 Riemann-Stieltjes Integral 214 -- Weights and Measures 214 -- Riemann-Stieltjes Integral 215 -- Space of Integrable Functions 221 -- Integrators of Bounded Variation 225 -- Riemann Integral 232 -- Riesz Representation Theorem 234 -- Other Definitions, Other Properties 239 -- 15 Fourier Series 244 -- Dirichlet's Formula 250 -- Fejer's Theorem 254 -- Complex Fourier Series 257 -- Part 3 Lebesgue Measure and Integration -- 16 Lebesgue Measure 263 -- Problem of Measure 263 -- Lebesgue Outer Measure 268 -- Riemann Integrability 274 -- Measurable Sets 277 -- Structure of Measurable Sets 283 -- A Nonmeasurable Set 289 -- Other Definitions 292 -- 17 Measurable Functions 296 -- Measurable Functions 296 -- Extended Real-Valued Functions 302 -- Sequences of Measurable Functions 304 -- Approximation of Measurable Functions 306 -- 18 Lebesgue Integral 312 -- Simple Functions 312 -- Nonnegative Functions 314 -- General Case 322 -- Lebesgue's Dominated Convergence Theorem 328 -- Approximation of Integrable Functions 333 -- 19 Additional Topics 337 -- Convergence in Measure 337 -- L[subscript p] Spaces 342 -- Approximation of L[subscript p] Functions 350 -- More on Fourier Series 352 -- 20 Differentiation 359 -- Lebesgue's Differentiation Theorem 359 -- Absolute Continuity 370 |
| Summary |
"Aimed at advanced undergraduates and beginning graduate students, Real Analysis offers a rigorous yet accessible course in the subject. Carothers, presupposing only a modest background in real analysis or advanced calculus, writes with an informal style and incorporates historical commentary as well as notes and references." "The book looks at metric and linear spaces, offering an introduction to general topology while emphasizing normed linear spaces. It addresses function spaces and provides familiar applications, such as the Weierstrass and Stone-Weierstrass approximation theorems, functions of bounded variation, Riemann-Stieltjes integration, and a brief introduction to Fourier analysis. Finally, it examines Lebesgue measure and integration on the line. Illustrations and abundant exercises round out the text." "Real Analysis will appeal to students in pure and applied mathematics as well as researchers in statistics, education, engineering, and economics."--Jacket |
| Bibliography |
Includes bibliographical references (pages 379-393) and index |
| Notes |
Print version record |
| Subject |
Mathematical analysis.
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MATHEMATICS -- Calculus.
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MATHEMATICS -- Mathematical Analysis.
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Mathematical analysis
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Reelle Analysis
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Lebesgue-integralen.
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Topologie.
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Lineaire algebra.
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Metrische ruimten.
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Functionaalanalyse.
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Análise real (análise)
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| Form |
Electronic book
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| ISBN |
9781139648714 |
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1139648713 |
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9780511814228 |
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0511814224 |
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9781139638265 |
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1139638262 |
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9781139641104 |
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1139641107 |
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